Properties

Label 2-966-1.1-c5-0-58
Degree $2$
Conductor $966$
Sign $-1$
Analytic cond. $154.930$
Root an. cond. $12.4471$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 9·3-s + 16·4-s − 54·5-s + 36·6-s + 49·7-s − 64·8-s + 81·9-s + 216·10-s + 564·11-s − 144·12-s + 476·13-s − 196·14-s + 486·15-s + 256·16-s − 1.30e3·17-s − 324·18-s + 14·19-s − 864·20-s − 441·21-s − 2.25e3·22-s + 529·23-s + 576·24-s − 209·25-s − 1.90e3·26-s − 729·27-s + 784·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.965·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.683·10-s + 1.40·11-s − 0.288·12-s + 0.781·13-s − 0.267·14-s + 0.557·15-s + 1/4·16-s − 1.09·17-s − 0.235·18-s + 0.00889·19-s − 0.482·20-s − 0.218·21-s − 0.993·22-s + 0.208·23-s + 0.204·24-s − 0.0668·25-s − 0.552·26-s − 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(154.930\)
Root analytic conductor: \(12.4471\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: $\chi_{966} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 966,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{2} T \)
3 \( 1 + p^{2} T \)
7 \( 1 - p^{2} T \)
23 \( 1 - p^{2} T \)
good5 \( 1 + 54 T + p^{5} T^{2} \)
11 \( 1 - 564 T + p^{5} T^{2} \)
13 \( 1 - 476 T + p^{5} T^{2} \)
17 \( 1 + 1308 T + p^{5} T^{2} \)
19 \( 1 - 14 T + p^{5} T^{2} \)
29 \( 1 + 6702 T + p^{5} T^{2} \)
31 \( 1 + 10258 T + p^{5} T^{2} \)
37 \( 1 - 14078 T + p^{5} T^{2} \)
41 \( 1 + 2826 T + p^{5} T^{2} \)
43 \( 1 - 14216 T + p^{5} T^{2} \)
47 \( 1 + 9990 T + p^{5} T^{2} \)
53 \( 1 - 16698 T + p^{5} T^{2} \)
59 \( 1 - 19044 T + p^{5} T^{2} \)
61 \( 1 - 5678 T + p^{5} T^{2} \)
67 \( 1 + 39736 T + p^{5} T^{2} \)
71 \( 1 - 60108 T + p^{5} T^{2} \)
73 \( 1 + 7714 T + p^{5} T^{2} \)
79 \( 1 - 106472 T + p^{5} T^{2} \)
83 \( 1 + 7662 T + p^{5} T^{2} \)
89 \( 1 + 76536 T + p^{5} T^{2} \)
97 \( 1 + 19048 T + p^{5} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.991323309486170526426405992931, −8.009519057128156191549241420301, −7.24322693916013387409876331047, −6.48507886507496221618110106853, −5.57224838337369285654630775865, −4.20507948091677939966079811735, −3.70312765697564789537030882182, −2.01370320555062652858954455067, −1.01704524024465910934758982293, 0, 1.01704524024465910934758982293, 2.01370320555062652858954455067, 3.70312765697564789537030882182, 4.20507948091677939966079811735, 5.57224838337369285654630775865, 6.48507886507496221618110106853, 7.24322693916013387409876331047, 8.009519057128156191549241420301, 8.991323309486170526426405992931

Graph of the $Z$-function along the critical line